The domain isn't really much of a problem, so long as the two bounding functions converge there. The sandwich principle says

If a function is bound between two convergent functions that converge to respectively then is somewhere between and (providing that is a convergent function). Hence, if then

Please observe that, if a function is bound between two functions that do not converge to the same limit, this does not mean that the first function converges.
Take, for example and . Clearly , and .
Just because is bound between two convergent series does not mean that it converges